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1002.3816v1 - A NOTE ON HYPERVECTOR SPACES...

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arXiv:1002.3816v1 [math.GM] 19 Feb 2010 A NOTE ON HYPERVECTOR SPACES SANJAY ROY , T. K. SAMANTA Department of Mathematics, South Bantra Ramkrishna Institution , India. Department of Mathematics, Uluberia College,West Bengal, India. E-mail : [email protected], mumpu [email protected] Abstract. The main aim of this paper is to generalize the concept vec- tor spaces by the hyperstructure. We generalize some definitions such as hypersubspaces, linera combination, Hamel basis, linearly dependence and linearly independence. A few important results like deletion theorem, exten- sion theorem, dimension theorem have been established in this hypervector space. Key Words : Hyperoperation, Hyperfield, Hypervector spaces, linearly dependent, linearly independent, basis in Hypervector space, deletion theo- rem, extension theorem, dimension of a hypervector space. 1. Introduction The concept of hyperstructure was first introduced by Marty [7] in 1934 at the 8th congress of scandinavian Mathematicians and then he established the definition of hypergroup [7] in 1935 to analysis their properties and ap- plied them to groups rational algebraic functions. Also he was motivated to introduced this structure to study several problems of the non-commutative algebra. Then several researchers have been worked on this new field of mod- ern algebra and developed it. M. Krasner [9], a great researcher in this area, introduced the notions of hyperring and hyperfield to use it as a technical tool in a study of his on the approximation of valued fields. Later on it has been developed and generalized by other researchers. Then the notion of the hypervector spaces was introduced by M. Scafati Tallini [10] in 2002. In the definition[10] of hypervector spaces, M.Scafati Tallini has considered the field as a usual field. In this paper, we have generalized the definition of hypervector space by considering the field as a hyperfield and also considering the structure multiplication of a vector by a scalar as hyperstructure. We again 1
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2 SANJAY ROY ,T. K. SAMANTA call it a hypervector space. Then we have established a few basic properties in this hypervector space and thereafter the notions of linear combinations, linearly dependence, linearly independence, Hamel basis, etc. are introduced and several important properties like deletion theorem, extension theorem etc. are developed. 2. Preliminaries We quote some definitions and proof of a few results which will be needed in the sequel. Definition 2.1. A hyperoperation over a non empty set X is a mapping of X × X into the set of all non empty subsets of X. Definition 2.2. A non empty set R with exactly one hyperoperation # is a hypergroupoid if x # y negationslash = Φ for all x , y in R. otherwise it is called a partial hypergroupoid. Let ( X , #) be a hypergroupoid. For every point x X and every non empty subset A of X , we defined x # A= uniontext { x # a } .
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